Problem: Simplify the following expression and state the condition under which the simplification is valid. $z = \dfrac{t^3 - 6t^2 - 27t}{9t^2 - 36t - 189}$
First factor out the greatest common factors in the numerator and in the denominator. $ z = \dfrac {t(t^2 - 6t - 27)} {9(t^2 - 4t - 21)} $ $ z = \dfrac{t}{9} \cdot \dfrac{t^2 - 6t - 27}{t^2 - 4t - 21} $ Next factor the numerator and denominator. $ z = \dfrac{t}{9} \cdot \dfrac{(t + 3)(t - 9)}{(t + 3)(t - 7)}$ Assuming $t \neq -3$ , we can cancel the $t + 3$ $ z = \dfrac{t}{9} \cdot \dfrac{t - 9}{t - 7}$ Therefore: $ z = \dfrac{ t(t - 9)}{ 9(t - 7)}$, $t \neq -3$